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The local bakery bakes more than a thousand 1- pound loaves of bread daily, and the weight of these loaves
varies. The mean weight is 1 Ib, and 1 oz., or 482 grams. Assume the standard deviation of the
weights is 18 grams and a sample of 40 loaves is to be randomly selected.
a) This sample of 40 has a mean value of x, which belongs to a sampling distribution.
Find the shape of this sampling distribution.
The sampling distribution shape depends on the original distribution, which is not stated, so the shape
can not be precisely determined. By the Central Limit Theorem, the shape of the mean
approaches the shape of a normal distribution as the sample size increases.
b) Find the mean of the sampling distribution.
482, same as the original distribution
c) Find the standard error of this sampling distribution,
18/sqrt(40) =
2.846
d) What is the probability that this sample mean will be between 475 and 495 grams?
P(475 < X < 495)
475 − 482
495 − 482
<Z<
)
2.846
2.846
= P(−2.460 < Z < 4.568)
= P(
= P( Z < 4.568) − P( Z < −2.460)
= 1.000 − 0.007
= 0.993
e) What is the probability that the sample mean will have a value less than 478 grams?
P ( X < 478)
478 − 482
= P( Z <
)
2.846
= P ( Z < −1.405)
= 0.080
f) What is the probability that the sample mean will be within 5 grams of the mean?
P(477 < X < 487)
477 − 482
487 − 482
<Z<
)
2.846
2.846
= P(−1.757 < Z < 1.757)
= P(
= P( Z < 1.757) − P( Z < −1.757)
= 0.9605 − 0.0395
= 0.921